Here is a collection of documents I have made over the years. I cannot claim that they are complete, neat, or even always correct, but they helped get me through a maths degree, and cover lots of interesting concepts.
Here is my masters dissertation:
The Inverse Galios Problem – The Rigidity Method – this gives an introduction to the inverse Galois problem, develops the relevant theory required to understand the rigidity method, a method that has been used to great effect in this area, and proves the rigitidy criterion. This takes up about the first half of the paper. The second half is a collection of numerous applications of this method culminating in one of the more celebrated results in the area; proof that the monster group can be realised as a Galois group!
Here is a collection of some notes I made for several courses during my undergrad at Imperial. These were typed up during the lectures so expect a few errors.
Riemann Geometry – the contents seems to have fallen off this document. It contains
- An introduction to manifolds
- Riemannian metrics and manifolds
- Jacobi fields
- Constant curvature
Groups and Representation Theory – this is very rough around the edges, with annotations and questions I had during the lectures dotted around. It remains a beautiful course however.
Elliptic Curves – covers the main concepts and results in this area. Introduces the p-adic numbers and considers elliptic curves over